2013 IdeaLab participants discuss cutting edge research ideas with both peers and leaders in their field

Interested in discussing cutting edge research ideas with both peers and leaders in their field?

Interested in broadening your professional network across the mathematical sciences?

Interested in the opportunity to present your ideas and hear about funding opportunities from program officers?

Idea-Lab is a one-week program aimed at early career researchers (within five years of their Ph.D.) that will focus on two different topics at
the frontier of research. Participants will be exposed to problems whose solution may require broad perspectives and multiple areas of expertise. Senior researchers will introduce the topics in
tutorials and lead discussions. The participants will break into teams to brainstorm ideas, comprehend the obstacles, and explore possible avenues towards solutions. The teams will be encouraged to develop a
research program proposal. On the last day, they will present their ideas to one another and to a small panel of representatives from funding agencies for feedback and advice.

IdeaLab applicants should be at an early stage of their post-Ph.D. career. A CV, research statement, and two reference letters are required.

Topics

The biological world at the scale of cellular organisms is full of fascinating examples of fluid motion that is generated or affected by its interaction with elastic structures.
Examples are the fluid motion around "swimming'' bacteria and sperm, and the ciliary function in the respiratory system.
A common feature of these phenomena is the interaction of elastic flexible membranes or filaments with a surrounding fluid, where the forces generated by the elastic
structures and their motion are coupled by the fluid dynamics.

The development of computational methods for the accurate simulation of thin filaments in a fluid has reached maturity.
At the same time, the force-generating mechanism of eukaryotic flagella and cilia has been well-studied biologically.
However, the vast majority of numerical models of flagellar and ciliary motions do not yet include a proper representation of the internal microtubule structure of flagella.

By bringing together mathematicians with a variety of backgrounds, the goal of this IdeaLab is to brainstorm on possible approaches to introduce a more
faithful representation of the internal structure of flagella into a computational model that can be used to study a variety of flows generated by microorganisms.

Simulation of flow streamlines generated by anorganism with one rotating flagellum bundle near a plane wall.

The collinear problem forms a planar slice of the full planar three-body problem. Figure courtesy of Rick Moeckel.

We will discuss several specific projects at the interface of mechanics, geometry and analysis.

The fascinating phenomenon of stabilization by vibration suggests one group of problems.
The most famous example of such a stabilization is the Kapitsa pendulum in which the upside-down
unstable equilibrium of the standard pendulum becomes a stable equilibrium when the pendulum's pivot is
vibrated vertically at a high enough frequency. See, for instance, the following YouTube video:

This effect led to the invention of the cyclotron and of the Paul trap, for which W. Paul received the 1989 Nobel Prize in physics.
Another effect in the same spirit is the stabilization of a viscous fluid by vibration.
A surface of molasses in an appropriately vibrating container can be made to form a vertical wall!
One proposed activity will be to recast these problems in terms of differential geometry, as the study of geodesics on
vibrating manifolds. This recasting has not been widely explored despite its important applications.

In celestial mechanics vibrational stablization questions also arise.
For example, the equal mass planar three-body problem (a three degree of freedom system after reductions) contains
four invariant submanifolds of codimension 2: the collinear three-body problem and its three isosceles sub-problems.
These four sub-problems are much better understood than the full planar problem. Oscillations orthogonal to their
submanifolds have the potential of connecting the submanifolds in interesting and not well-understood ways.
Or perhaps such connections are blocked in some way. From the perspective of differential geometry, we have a
Riemannian three-manifold which contains 4 distinguished totally geodesic surfaces.
What can we say about the growth or oscillation of the normal mode (orthogonal to the surface)
of the Jacobi equation for geodesics lying in one of these surfaces? How does this understanding of
normal modes lead to a better understanding of the full geodesic flow?

The discussion will greatly benefit from collaboration of people with diverse interests ranging from
geometry to differential equations to mechanics.